Question

If $f\left(x\right)={\log }_{[x-1]}\left(\frac{|x|}{x}\right),$ where $[\cdot ]$ is greatest integer function, then domain $D_f$ and range $R_f$ of $f\left(x\right)$ are

• A. $D_f=[3,\infty )$
• B. $R_f=\left\{0\right\}$
• C. $R_f=R$
• D. $D_f=R$

Answer 1

Answer: (A), (B)

The correct options are
A $D_f=[3,\infty )$
B $R_f=\left\{0\right\}$

For $f\left(x\right)$ to be defined
$[x-1]>0,[x-1]\ne 1$ and $\frac{|x|}{x}>0$ Now ,$[x-1]>0,[x-1]\ne 1\Rightarrow x\in [3,\infty )$
and, $\frac{|x|}{x}>0\Rightarrow x>0$
So we have domain $D_f\in [3,\infty )$
Now, $\forall x\in [3,\infty ),\frac{|x|}{x}=1$
$\therefore f\left(x\right)={\log }_{[x-1]}\left(\frac{|x|}{x}\right)={\log }_{[x-1]}1=0$
So, $R_f=\left\{0\right\}$